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Mirrors > Home > MPE Home > Th. List > wunfv | Structured version Visualization version GIF version |
Description: A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunfv | ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | 1, 2 | wunrn 9430 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
4 | 1, 3 | wununi 9407 | . 2 ⊢ (𝜑 → ∪ ran 𝐴 ∈ 𝑈) |
5 | fvssunirn 6127 | . . 3 ⊢ (𝐴‘𝐵) ⊆ ∪ ran 𝐴 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴‘𝐵) ⊆ ∪ ran 𝐴) |
7 | 1, 4, 6 | wunss 9413 | 1 ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ran crn 5039 ‘cfv 5804 WUnicwun 9401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-cnv 5046 df-dm 5048 df-rn 5049 df-iota 5768 df-fv 5812 df-wun 9403 |
This theorem is referenced by: (None) |
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